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The prime example I can think of is Russia. In 1917, the February Revolution put a coalition government, headed by Alexander Kerensky, in charge after Tsar Nicolas abdicated the throne. The October Revolution put Lenin and the communists in charge of the government although a civil war and unsuccessful counter revolutions went on until 1921.

Edit to add some additional ones, based on the criteria of less than 12 months between coups...

In Bolivia, on October 6, 1970 Juan José Torres seized power from Alfredo Ovando Candía. In turn, on August 21, 1971, General Hugo Banzer staged a successful coup with the help of the military and, perhaps, assistance as part of Operation Condor.

In Burundi in July 1966, Prince Ntare V deposed his father. On November 28th of 1966, Michel Micombero, who had earlier assisted Ntare, staged a coup, abolished the monarchy and took power himself.

In Ethiopia, September 12, 1974 Aman Mikael Andom overthrew Emperor Haile Selassie I. On November 17th 1974, Andom was killed by forces loyal to the Provisional Military Administrative Council (aka Derg).

Haiti saw several coups from 1988 onward after the departure of Jean-Claude "Baby Doc" Duvalier in 1986 up until the threat of US intervention in 1994. Leadership changed hands several times and exactly who was in charge wasn't always clear.

In 1994, the Rwandan Civil War resulted in several leadership changes and terrible genocide.

There are certainly some others I didn't cover here so feel free to add more examples.

In 251 BC, Nicocles murdered Paseas to become the ruler of Sicyon. Four months later, he was deposed by a group of exiles led by Aratus.

In 68 AD, the Roman Senate and army transferred their allegiance from Nero to Galba. In 69, the Year of the Four Emperors, Galba was assassinated by troops bribed by Otho, then Otho lost in battle to Vitellius and killed himself, then Vitellius was killed by solders loyal to Vespasian.

In 192 AD, China was ruled by a puppet Emperor with effective power in the hands of Dong Zhuo. Dong was killed by his adopted son Lü Bu, with Wang Yun then heading the government. Wang did not last long, as Dong's top generals retook the capital, driving out Lü Bu and executing Wang Yun.

On the last day of 192 AD, Emperor Commodus was killed. Pertinax was appointed Emperor the next day, but lasted three months before being killed by soldiers who then sold the imperial position to Didius Julianus. Septimius Severus arrived at the head of his provincial army two months later and he then became Emperor.

In 198 AD, the warlord Zhang Yang was planning to send troops to assist Lü Bu against Cao Cao when he was assassinated by his general Yang Chou, who intended instead to join Cao. Yang was then assassinated by Sui Gu, who took command of the troops before being defeated and killed by Cao's general Shi Huan.

The number of such cases is so high that it would be difficult to put all them here.

Consider the following:

Iraq's Abdul Rahman Arif was a key figure behind the coup of 1963 after which he became the commander in chief and his brother the president. After the 1966 death of his brother he became the president of Iraq and was overthrown in 17 July Revolution in 1968 by the Baath party.

Cuba's Fulgencio Batista came to power in a military coup in 1952 and was overthrown by Fidel Castro in 1959

Argentina's Arturo Rawson ruled only three days between coups in 1943.

Argentina's Jorge Rafael Videla came to power in a coup of 1976 and was disposed and arrested in 1981. His successor Eduardo Viola was disposed by a coup less than a year later.